| L(s) = 1 | − 2.38·2-s + 2.75·3-s + 3.70·4-s − 0.982·5-s − 6.57·6-s − 4.06·8-s + 4.57·9-s + 2.34·10-s − 0.587·11-s + 10.1·12-s − 2.70·15-s + 2.30·16-s + 6.45·17-s − 10.9·18-s + 3.82·19-s − 3.63·20-s + 1.40·22-s + 8.26·23-s − 11.1·24-s − 4.03·25-s + 4.32·27-s − 3.96·29-s + 6.45·30-s + 2.98·31-s + 2.62·32-s − 1.61·33-s − 15.4·34-s + ⋯ |
| L(s) = 1 | − 1.68·2-s + 1.58·3-s + 1.85·4-s − 0.439·5-s − 2.68·6-s − 1.43·8-s + 1.52·9-s + 0.741·10-s − 0.177·11-s + 2.94·12-s − 0.697·15-s + 0.576·16-s + 1.56·17-s − 2.57·18-s + 0.877·19-s − 0.813·20-s + 0.299·22-s + 1.72·23-s − 2.28·24-s − 0.807·25-s + 0.831·27-s − 0.735·29-s + 1.17·30-s + 0.536·31-s + 0.464·32-s − 0.281·33-s − 2.64·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.730641978\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.730641978\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.38T + 2T^{2} \) |
| 3 | \( 1 - 2.75T + 3T^{2} \) |
| 5 | \( 1 + 0.982T + 5T^{2} \) |
| 11 | \( 1 + 0.587T + 11T^{2} \) |
| 17 | \( 1 - 6.45T + 17T^{2} \) |
| 19 | \( 1 - 3.82T + 19T^{2} \) |
| 23 | \( 1 - 8.26T + 23T^{2} \) |
| 29 | \( 1 + 3.96T + 29T^{2} \) |
| 31 | \( 1 - 2.98T + 31T^{2} \) |
| 37 | \( 1 - 1.75T + 37T^{2} \) |
| 41 | \( 1 + 3.67T + 41T^{2} \) |
| 43 | \( 1 - 6.38T + 43T^{2} \) |
| 47 | \( 1 - 4.34T + 47T^{2} \) |
| 53 | \( 1 - 0.425T + 53T^{2} \) |
| 59 | \( 1 + 6.00T + 59T^{2} \) |
| 61 | \( 1 + 2.20T + 61T^{2} \) |
| 67 | \( 1 - 7.01T + 67T^{2} \) |
| 71 | \( 1 - 3.60T + 71T^{2} \) |
| 73 | \( 1 + 4.93T + 73T^{2} \) |
| 79 | \( 1 - 2.78T + 79T^{2} \) |
| 83 | \( 1 - 2.86T + 83T^{2} \) |
| 89 | \( 1 - 2.09T + 89T^{2} \) |
| 97 | \( 1 + 7.69T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.923187623193699623279492344970, −7.45209807045930139858836208840, −7.13904796635755634797440280500, −5.99390335732060589672541241665, −4.99832945572756562393241745975, −3.84779128382525417452128166393, −3.15904420241212082087027509835, −2.57121906839704450433606869481, −1.58866920568884581668572166523, −0.818696689497890583894886432251,
0.818696689497890583894886432251, 1.58866920568884581668572166523, 2.57121906839704450433606869481, 3.15904420241212082087027509835, 3.84779128382525417452128166393, 4.99832945572756562393241745975, 5.99390335732060589672541241665, 7.13904796635755634797440280500, 7.45209807045930139858836208840, 7.923187623193699623279492344970
